Product rule

If a task can be broken into two stages, with $m$ ways to do the first stage and $n$ ways to do the second stage (for each way of doing the first), then there are $m \times n$ total ways to complete the task. This generalizes to more stages: if there are $k$ stages with $n_1, n_2, ..., n_k$ ways at each stage, the total number of ways is $n_1 \times n_2 \times \cdots \times n_k$.

Important properties

• The product rule applies only when choices at each stage are independent.

• It can be extended to any number of stages.

• It is fundamental for counting sequences, arrangements, and ordered selections.